Filippos Tzortzakakis, Howard Community College
Burhanuddin Marvi, Howard Community College
Mentored by Alex M. Barr, Ph.D.
The detection of gravitational waves has commenced a new era of Astronomy. This paper will go into detail about the first ever directly detected Gravitational Wave event (GW150914); a merging of a binary black hole system. For this event, data obtained from the Laser Interferometer Gravitational-Wave Observatory (LIGO) is analyzed to reproduce the official findings, utilizing only introductory physics concepts and a spreadsheet computer program. Specifically, focus is given to determine the mass of each of the two black holes, the distance of the event from the Earth, and the energy emitted in the form of gravitational waves. The results acquired are in general agreement with the LIGO findings, as all of them have an acceptable error with the largest being a factor of 15. Accurate results are achieved without complex mathematical formulas or a detailed knowledge of general relativity.
What are Gravitational Waves?
Gravitational waves are ripples in the fabric of spacetime. Every mass in the universe creates an indentation in space itself . To better understand this indentation, imagine the universe as a stretchy cloth. One could envision planets and stars as marbles proportionate to their actual size and mass. When one places a smaller marble on the cloth, the cloth bends slightly downward because it cannot fully support the mass of the marble. Now, if a marble is placed into orbit, one can imagine the cloth oscillating as the mass travels and disrupts different portions of the surface of the fabric. This analogy can also be applied to our universe. Larger masses such as the sun create even bigger indentations in spacetime. Ripples due to gravitational waves or indentation caused by a substantial mass can stretch/distort the fabric of spacetime the same way the cloth is being stretched by the marble. These indentations play a vital role in how masses in space orbit each other. For example, the Earth orbits the sun in an ellipse meaning that velocity is changing, therefore the Earth accelerates. This acceleration is important because gravitational waves can only be produced by an accelerating mass. An analogous requirement is seen in electromagnetism, where only accelerating charges produce electromagnetic radiation. Although the Earth does technically produce gravitational waves, they are undetectable because its mass is not great enough. Most detectable gravitational waves come from black holes and neutron stars. These celestial objects are truly massive.
The GW150914 Detection
In this paper, we will be talking about the first gravitational wave detection GW150914 made on September 14, 2015. This event involved the merger of two black holes of 36 and 29 solar masses, around six sextillion (6 x 1021) miles away from Earth . Before the merger, the two black holes were orbiting each other, while the radius of their orbit kept decreasing. As a result, the black holes were continuously accelerating and eventually merged. During the merger, speeds close to the speed of light were achieved and the gravitational waves emitted had the greatest amplitude.
The first gravitational wave detection was made by the Laser Interferometer Gravitational-Wave Observatory (LIGO). LIGO is a National Science Foundation (NSF) funded project with two facilities: one in Livingston, Louisiana and one in Hanford, Washington. Its goal is to detect gravitational waves with the use of laser interferometers. Interferometers are apparatuses that merge two sources of light together to produce an interference pattern . The LIGO team used a Michelson interferometer, in which a laser beam is split into two identical beams at a 90-degree angle. Each beam travels through a 2.5-mile-long tunnel, and when it reaches its end, it is reflected by a mirror. After several trips back and forth through their respective tunnels, the two beams return to the point where they were initially split and merge again, generating an interference pattern.
If the two beams do not merge at the same time, and therefore create a shift in the interference pattern, the only conclusion would be that one beam travelled a longer distance inside its tube. That indicates that the length of one arm/tunnel had to increase. When gravitational waves pass through the facility, they cause the fabric of spacetime to elongate in one direction and shorten in the perpendicular direction. Thus, one arm stretches while the other shrinks and that is what causes one of the lasers to travel a longer distance than the other.
These interferometers must be extremely sensitive to pick up the slightest change in the distance a laser travels. This sensitivity means that most of the data that LIGO collects is not from gravitational waves. Vibrations from oncoming vehicles, storms, and earth movements can cause the device to generate false readings. LIGO has taken precautions to filter out this “bad” data by cross-referencing the waves generated in both of their facilities . For instance, a wave passing through the Louisiana facility would produce a certain waveform, and if it is a genuine gravitational wave, not a passing vehicle or a storm, the same wave would produce an analogous waveform at the Washington facility a specific time interval later.
Strain and its Importance
As mentioned above, gravitational waves can inflict strain on spacetime itself. Strain is the measurement of elongation or shortening of an object’s dimensions based on its initial dimensions. The dimensions of an object increase and decrease repeatedly based on the strain and frequency of the gravitational wave. Since strain is the change in length over an initial length, it is unitless. Strain is crucial because scientists are only able to obtain strain data through the interference pattern given by the interferometer. All other elements of the gravitational wave, such as its frequency, can only be derived from the strain data collected.
For this project, a strain vs time table was obtained from the LIGO Open Science Center website . We chose to analyze the simulated signal data which is less noisy than the actual observational data. All the findings in this paper are to be compared with the ones from the official LIGO paper released in Physical Review Letters .
Our work was greatly aided by a publication by Lior M. Burko which outlined many of the calculations needed to acquire findings like LIGO . It is vital to note that gravitational waves behave as a normal wave, described in the following wave equation ,
where h is strain, A the amplitude and ω the angular frequency of the wave.
This equation will be used to obtain the angular frequency and frequency in Section I. Then, in Section II, the chirp mass and the masses of the two black holes will be calculated. Finally, the distance of the event from the Earth and the total energy emitted via gravitational waves will be found in Section III.
Section I: Frequency Calculations
As mentioned, the data provided by LIGO is a strain vs time table. By using these data to plot a strain vs time graph, found in Figure 1, a simple waveform will appear, where as time passes, its frequency and amplitude increases due to the merging. It must be noted that the time is defined such that the merging occurs at t = 0. The square of the angular frequency of the gravitational wave can be calculated by taking the negative second time derivative of the strain (equation (1)) and dividing by the strain. This happens because the sines cancel out.
Figure 1: Waveform for the gravitational wave produced by the merger of two black holes. The black holes spiral inward before merging together at t = 0.
Having the value of the square of the angular frequency at each moment in time, it is easy to also calculate the frequency for each time value. The following equation can be used for that conversion.
A graph of frequency vs time is shown in Figure 2. The frequency of the gravitational waves kept increasing, meaning the two black holes were getting closer to each other.
Figure 2: As the black holes spiral towards each other their orbital speeds increase resulting in an increasing frequency of gravitational waves emitted.
Initially, the frequency vs time graph had a lot of noise. To minimize its noise and produce a smoother scatter plot, an averaging technique for the frequency values was vital. For smoothing the frequency vs time graph, the method applied was averaging values which were later used in calculating frequency. The first 50 values of were averaged into one value thus replacing the first value with that average. Then another 50 values, starting from the second value were averaged and again replacing the second value. Each 50 values correspond to a time interval just over 0.01 seconds so the time over which the averaging is done is quite small. This process of averaging, referred to as “moving average” in time series analysis, was repeated to replace all values of and, therefore, all frequency values too.
Section II: Mass Calculations
Another important quantity needed is the chirp mass of the binary black hole system. The chirp mass depends on the masses of the two black holes and can therefore be used to determine their mass. From Section I, the frequency of the gravitational wave is known, so its derivatives can also be calculated. Rearranging equation (2) in Lior’s M. Burko paper, the chirp mass of the system is given by,
where c is the speed of light, f is the frequency of the gravitational wave, is the first time derivative of the frequency and G is the gravitational constant.
Having the chirp mass values for each time value, a graph can be produced. Unfortunately, the numerical derivatives of frequency did not provide a clean graph for the chirp mass, so an analytical expression for frequency was needed. A curve fit to the frequency vs time graph yielded an analytical expression for the frequency,
which was used to calculate the values needed for the chirp mass. The resulting graph of chirp mass vs time is found in Figure 3.
Figure 3: The chirp mass of the binary black hole system remains approximately constant until just before the black holes merge at t = 0. The sudden dip near t = -0.2 s is unexpected and likely the result of noise in the data.
Chirp mass is a combination of the two individual masses, which is essential for many of the calculations but does not have a specific physical interpretation. Using the chirp mass, the mass of each black hole can be calculated through equation (3), found in Lior’s M. Burko paper . For this paper, the masses of both black holes are taken to be equal, so that the mass can be found immediately with equation (3).
We find that the masses were approximately 36 M☉ each (M☉ being solar mass, ). A comparison with LIGO shows that the actual black hole masses were approximately 36 and 29 M☉ .
Section III: Distance and Energy Calculations
One of the most curious parts of analyzing the data was to find a way to calculate our distance from the event. In reference , one of the strain equations that included the distance from the event was incorrect because the units did not fully cancel other out. This is important because strain, by definition, is unitless. The correct equation for calculating the distance is
This equation was found in reference . For this equation, h is not the strain for each time value, but the amplitude of the strain wave, which can be found in Figure 1. The distance from Earth to the event was calculated to be approximately 909 megaparsecs. The estimated distance from LIGO was between 230-570 megaparsecs .
Finally, one would think that the merging of the two black holes would yield a single larger mass by adding the two products. However, this is not exactly accurate because during the merging energy – equivalent to mass – is lost and released in the form of gravitational waves. To determine this energy, one must first calculate the distance between the two black holes, just as the merging commences. The Schwarzschild radius is defined as the radius surrounding a nonrotating black hole . When two black holes are to be merged, the radius of each black hole starts overlapping each other. The formula for calculating the Schwarzschild radius was found in reference  and its expression is
Now having the Schwarzschild radius, the rate at which the energy is being lost to gravitational waves can be found. Using equation (12) from Hilborn’s paper for the power radiated through gravitational waves  and altering it to fit this paper’s notation,
To convert this into a value of energy, one must multiply the P expression by the time interval of two adjacent time values. By doing that, the energy loss for each small-time interval can be found. The data includes 2543 time intervals and thus 2543 values of energy lost to gravitational waves. Adding all these energy values together gives a total energy released of approximately 3.1*1046 J. Using Einstein’s equation the energy lost can be converted into solar masses. The mass lost was found to be equal to approximately 0.17 M☉. LIGO approximated the energy lost to be equal to 2.5-3.5 M☉.
Overall, our results were consistent with those of LIGO. The mass of each black hole was found to be 36 M☉, while LIGO found the black holes to be 29 and 36 M☉. Once again, we assumed each black hole equal in mass to simplify the math. Another important result was the distance calculation to find how far the event was from Earth. We found it to be 909 megaparsecs (1 megaparsec = 1.917 x 1019 miles), which is really close to LIGO’s approximation of 230-570 megaparsecs. There is no clear reason why our result is not in an even closer agreement with LIGO’s result. Nevertheless, no complete agreement could have been achieved since we utilized approximate formulas to obtain results.
Another significant calculation was the energy output in the form of gravitational waves. We found the energy lost to be around 0.17 M☉, while LIGO calculated it to be 2.5-3.5 M☉. We believe this underestimation is due to not taken special relativity into consideration. During the final moments of the merging, both black holes achieve speeds close to the speed of light. As any mass approaches the speed of light, the relativistic mass of an object will increase. Subsequently, a higher mass translates to more intense gravitational waves and more energy released through gravitational waves. However, despite using simplified formulas and not taking special relativity into account, all our results yield an acceptable magnitude of error, demonstrating that with basic knowledge of physics, one can conduct a surprisingly adequate analysis of a gravitational wave event.
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